Is my approach to building an RNN to predict the probability that the word is in English appropriate?
while using a neural network for this type of problem is not the ideal use-case, it is a good exercise.
In terms of conceptual issues, the most concerning that I see is the loss: $\sum_{i=1}^N L_i$.
First issue, is that it validates loss at each time step equivalently. This is probably not ideal because in the example (cat), we dont expect it to know its English from just c or ca, but at cat. The quickest fix and probably the best, is to just use $L = L_N$. Though an argument would be that the model should become more and more aware as it gets more letters, and this loss function doesnt achieve that, so another solution would be to add another fixed parameter that you can play with: $L = \sum_i r^{-i}L_i$ where $0 \lt r \lt 1$. Note that $r^{-i}$ can be replaced with any function that increases in size.
Another issue with the loss is it doesnt normalize, meaning on average, larger words will hold more weight to the model than smaller ones, while this may be intended it should be noted and considered (also note if you do end up going with $L=L_N$ this will no longer be a concern.
Hope this helps